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Spectral Graph Theory

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Spectral Graph Theory Spectral graph theory Uri Feige January 2010 1 Background With every graph (or digraph) one can associate several di®erent matrices. We have already seen the vertex-edge incidence matrix, the Laplacian and the adjacency matrix of a graph. Here we shall concentrate mainly on the adjacency matrix of (undirected) graphs, and also discuss briefly the Laplacian. We shall show that spectral properies (the eigenvalues and eigenvectors) of these matrices provide useful information about the structure of the graph. It turns out that for regular graphs, the information one can deduce from one matrix representation (e.g., the adjacency matrix) is similar to the information one can deduce from other representations (such as the Laplacian). We remark that for nonregular graphs, this is not the case, and the choice of matrix representation may make a signi¯cant di®erence. We shall not elaborate on this issue further, as our main concern here will be either with regular or nearly regular graph. The adjacency matrix of a connected undirected graph is nonnegative, symmetric and irreducible (namely, it cannot be decomposed into two diagonal blocks and two o®-diagonal blocks, one of which is all-0). As such, standard results n linear algebra, including the Perron-Frobenius theorem, imply that: 1. All its eigenvalues are real. Let us denote them by ¸1 ¸ ¸2 ::: ¸ ¸n. (Equality between eigenvalues corresponds to eigenvalue multiplicity.) 2. Eigenvectors that correspond to di®erent eigenvalues are orthogonal to each other. 3. The eigenvector that corresponds ¸1 is all positive. 4. ¸1 > ¸2 and ¸1 ¸ j¸nj. An eigenvector with eigenvalue ¸ can be interpreted as associating values (the coordinate entries of the eigenvalue) with the vertices of the graph, such that each vertex has a value that is the sum of its neighbors, scaled by 1=¸. P Observe that the trace of the adjacency matrix is 0 and hence ¸i = 0, implying in particular that ¸n < 0. There is useful characterization of eigenvalues by Raleigh quotients. Let v1; : : : ; vn be an orthonormal basis of eigenvalues. For a nonzero vector x, let a =< x; v > and hence P i i x = aivi. Observe that: 1 P xtAx ¸ (a )2 P i i t = 2 x x (ai) xtAx xtAx This implies that ¸n · xtx · ¸1. Moreover, if x is orthogonal to v1 then xtx · ¸2. For a d-regular graph, ¸1 = d, and the corresponding eigenvector is the all 1 vector. If the graph is disconnected, then its adjacency matrix decomposes into adjacency matrices of its connected components. In this case (and again, we assume here d-regularity), the multiplicity of the eigenvalue d is equal to the number of connected components. This indicates that a small gap between ¸1 and ¸2 corresponds to the graph having small cuts. There is also a converse, saying that a large gap between ¸1 and ¸2 implies that the graph has no small cuts in some exact sense. Namely, it is an expander. To understand intuitively this expansion property (in fact, a related property as we shall also consider ¸n here), consider a random walk starting at an arbitrary distribution x. After one step, the distribution is Ax (if G is regular) normalized to 1. By representing x in the basis of eigenvectors, and assuming that ¸1 is much larger (say, twice as large) than max[¸2; ¡¸n] it is clear that in O(log n) steps, the walk mixes. Hence there cannot be any \bottlenecks" (sets of vertices with only few outgoing edges), as then the walk would not mix. If the graph is not regular, the largest eigenvalue is at least as large as the average degree (by taking Raleigh quotient for all 1 vector) and not larger that the maximum value c for which the graph has a c-core (by sorting according to largest eigenvaluep - no vertex has more than ¸1 higher neighbors). For a star, the largest eigenvalue is n ¡ 1. If the graph is bipartite, nonzero eigenvalues come in pairs (flipping the entries of vertices of one side in the eigenvectors). Hence ¸n = ¡¸1. If j¸nj is much smaller than ¸1, this implies that the graph does not have very large cuts (that cut almost all the edges). The converse is not true, in the sense that j¸nj might be close to ¸1, and still the graph has no large cuts. This may happen for example if a d-regular graph G contains a small subgraph that is nearly a complete bipartite graph on 2d vertices. This small subgraph a®ects ¸n (as can be seen by taking Rayleigh quotients) but is insigni¯cant in terms of the existence of large cuts in G. Extensions of the eigenvalue based technique to semide¯nite programming (a topic beyond the scope of this course) give an approximation algorithm for max-cut with a very good approximation ratio (roughly 0.87856). The following theorem is due to Ho®man, and illustrates the connection between eigen- values and graph properties. Theorem 1 Let G be a d-regular n-vertex graph and let ¸n be the most negative eigenvalue of its adjacency matrix A. Then the size of its largest independent is at most n¸ ®(G) ·¡ n d ¡ ¸n Proof: Let S be an independent set in G. Consider the vector x with value n ¡ jSj on S and value ¡jSj on the remaining vertices. Then we use the Rayleigh quotient to bound ¸n from above. xtAx jSj2(nd ¡ 2jSjd) ¡ 2djSj(n ¡ jSj)jSj ¡njSj2d ¡jSjd = = = xtx jSj(n ¡ jSj)2 + (n ¡ jSj)jSj2 njSj(n ¡ jSj n ¡ jSj 2 The proof follows by rearranging. 2 2 2 2 P 2 P The eigenvalues of A are (¸i) . The trace of A implies that (¸i) = di. For pd-regular graphs, this implies that the average absolute value of eigenvalues is at most d. For random regular graphs, or random graphs that are nearly regular, it is in fact the case that with highp probability over the choice of graph, all but the largest eigenvalue have absolute value O( d). This can be proved by the trace method (considering higher even powers of A). An alternative proof can be based on considering the form xtAx for all possible unit vectors x orthogonal to v1. (There are in¯nitely many such vectors so a certain discretization needs to be used.) If G is regular or nearly regular, the adjacency matrix is most helpful. If the graph is irregular, the Laplacian may be more informative. Note that for regular graphs, the spectra of these two matrices are easily related. For general graphs, the Laplacian gives the P 2 quadratic form (i;j)2E(xi ¡ xj) . Hence it is positive semide¯nite, with the all 1 vector as a unique eigenvector of eigenvalue 0 (if the graph is connected). The next eigenvalue is known as the Fiedler value and the associated eigenvector as the Fiedler vector. It gives a useful heuristic for spectral graph partitioning. The Fiedler vector and the next eigenvector form a convenient coordinate system for graph drawing (see [3]). Much of what we said about the spectrum of regular graphs applies to graphs that are nearly regular. In establishing that a random graph in nearly regular, it is useful to have some bounds of the probability that a degree of a vertex deviates from its expectation. This bounds are quantitative versions of the law of large numbers and of the central limit theorem (that says that sums of independent random variables converge to the normal distribution), and we present one representative such bound. p1+:::+pn Theorem 2 Let p1; : : : ; pn 2 [0; 1] and set p = n . Assume that X1;:::Xn are mutually independent random variables with Pr[Xi = 1¡pi] = pi and Pr[Xi = ¡pi] = 1¡pi. Let X = X1 + ::: + Xn. (Observe that E[Xi] = 0, and E[X] = 0.) Then 2 3 2 Pr[X > a] < e¡a =2pn+a =2(pn) and 2 Pr[X < ¡a] < e¡a =2pn The theorem above is typically used when p · 1=2. Note that the bounds for X < ¡a seem stronger than those for X > a. They can be used for X > a after replacing p by 1 ¡ p. Namely, Pr[X > a] < e¡a2=2(1¡p)n. Note also that the bounds can be tightened in certain cases. See [1] for more details. 2 Refuting random 4SAT We illustrate here an application (due to Goerdt and Krivelevich [?]) of spectral techniques to a seemingly unrelated problem. A 4CNF formula contains clauses, where each clause contains four literals, and each literal is either a Boolean variable or its negation. An assignment satis¯es the formula if in each clause at least one literal is set to true. Checking whether a given 4CNF formula 3 is satis¯able is NP-complete. Namely, it is in NP (if the answer is yes there is a satisfying assignment certifying this), and it is NP-hard. Checking whether a given 4CNF formula is not satis¯able is coNP-complete. Hence there are no witnesses for non-satis¯ability unless coNP=NP. This applies for worst case instances. Here we study the average case (for some natural distribution). Consider the distribution Fn;m of random 4CNF formulas with n variables and m clauses. The formula is random in the sense that every literal in the formula is chosen independently at random. (We may further require that clauses are distinct and that variables in a clause are distinct, but this has negligible e®ect on the results that we present.) Observe that given any assignment to the variables, a random clause is satis¯ed by this assignment with probability exactly 15/16 (every literal has probability 1/2 of getting polarity opposite to the assignment).
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